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Re: Options for Eigenvalues

  • To: mathgroup at smc.vnet.net
  • Subject: [mg96145] Re: Options for Eigenvalues
  • From: Roman <rschmied at gmail.com>
  • Date: Sat, 7 Feb 2009 03:35:27 -0500 (EST)
  • References: <gmecfo$adp$1@smc.vnet.net>

Sebastian,

I'm parsing the error messages of Mathematica 6 here. Maybe there are
further undocumented methods and options.

The method should be either Automatic or Arnoldi.

A call should be formed, for example, like this:

Eigenvalues[H, 1, Method -> {Arnoldi, MaxIterations -> 10000, Criteria
-> RealPart}]

the extra options are:
   Shift (a complex number)
   Tolerance (non-negative real number or Automatic)
   BasisSize (integer >2 and <= matrix size)
   MaxIterations (integer)
   StartingVector (a vector of length equal to the matrix size)
   Criteria: can be
      Magnitude
      RealPart
      ImaginaryPart
      BothEnds (for real symmetric eigenvalue problems only)

To know what these mean, I recommend the ARPACK documentation:
http://www.caam.rice.edu/software/ARPACK/
Briefly, the basis size option gives the number of vectors used in the
iteration; not even the authors of ARPACK seem to know an optimal
choice here. The criteria specify which n eigenvalues are to be
computed: those with largest magnitude, largest real part, or largest
imaginary part. "BothEnds" computes alternating eigenvalues with small
and large real parts. "Shift" is for shifting the spectrum around
(shift-invert method) to pick out eigenvalues from the center of the
spectrum, close to a previously known value.

hth
Roman.


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